Finite Group Databasebeta

← C7×ΓL(1,𝔽8)×D75 #36347754

Subgroups — C7×ΓL(1,𝔽8)×D75 #36347754

Order 22050.

132 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×ΓL(1,𝔽8)×C25C211yesmaximal, characteristic, complemented, Hall
2C72×D75C311yesmaximal, characteristic, complemented
3C7×ΓL(1,𝔽8)×D25—31nomaximal
4C7×ΓL(1,𝔽8)×D15—51nomaximal
5C7×ΓL(1,𝔽8)×C25—21no—
6C7×ΓL(1,𝔽8)×C25D311yescharacteristic, complemented
7C3×C72×C25C2×C311yescharacteristic, complemented
8C3×C7×D75—71nomaximal
9ΓL(1,𝔽8)×D75C711yesmaximal, characteristic, direct product factor
10C72×D25—31noHall
11C3×C5×C7×ΓL(1,𝔽8)D511yescharacteristic
12C32×C7×C25—71no—
13C3×ΓL(1,𝔽8)×C25C2×C711yescharacteristic, complemented
14C72×D15—51no—
15C7×D5×ΓL(1,𝔽8)—151no—
16C72×C25C3×D311yescharacteristic, complemented, Hall
17C7×D75—32no—
18C7×D75ΓL(1,𝔽8)11yescharacteristic, direct product factor
19C7×D75C3×C711yescharacteristic, complemented
20C3×C7×D25—211no—
21ΓL(1,𝔽8)×D25—31no—
22D3×C7×ΓL(1,𝔽8)—251noHall
23C3×C5×C72C3×D511yescharacteristic
24C5×C7×ΓL(1,𝔽8)—21no—
25C5×C7×ΓL(1,𝔽8)D1511yescharacteristic
26C3×C7×D15—351no—
27ΓL(1,𝔽8)×D15—51no—
28C3×C7×C25—32no—
29C3×C7×C25—71no—
30C3×C7×C25—141no—
31C3×C7×C25C2×ΓL(1,𝔽8)11yescharacteristic, complemented
32C3×C7×C25C2×C3×C711yescharacteristic, complemented
33ΓL(1,𝔽8)×C25—21no—
34ΓL(1,𝔽8)×C25D3×C711yescharacteristic, complemented
35C72×D5—151no—
36C3×D75—71noHall
37C3×C7×ΓL(1,𝔽8)D2511yescharacteristic, complemented, Hall
38C7×D25—31no—
39C7×D25—31no—
40C7×D25—92no—
41C32×C5×C7—71no—
42C3×C5×ΓL(1,𝔽8)C7×D511yescharacteristic
43D3×C72—251no—
44C2×C7×ΓL(1,𝔽8)—751no—
45C5×C72C3×D1511yescharacteristic
46C32×C25—71noHall
47C7×D15—51no—
48C7×D15—51no—
49C7×D15—152no—
50C3×C7×D5—1051no—
51D5×ΓL(1,𝔽8)—151no—
52C7×C25—32no—
53C7×C25D3×ΓL(1,𝔽8)11yescharacteristic, complemented
54C7×C25C3×D3×C711yescharacteristic, complemented
55D75C7×ΓL(1,𝔽8)11yescharacteristic, direct product factor
56C3×D25—211no—
57C3×C72C3×D2511yescharacteristic, complemented
58C7×ΓL(1,𝔽8)—21no—
59C7×ΓL(1,𝔽8)D7511yescharacteristic, direct product factor
60C3×D3×C7—1751no—
61D3×ΓL(1,𝔽8)—251no—
62C3×C5×C7—32no—
63C3×C5×C7—71no—
64C3×C5×C7—141no—
65C3×C5×C7D5×ΓL(1,𝔽8)11yescharacteristic
66C3×C5×C7C3×C7×D511yescharacteristic
67C5×ΓL(1,𝔽8)—21no—
68C5×ΓL(1,𝔽8)C7×D1511yescharacteristic
69C2×C72—751noHall
70C3×D15—351no—
71C3×C25—71no—
72C3×C25—141no—
73C3×C25C2×C7×ΓL(1,𝔽8)11yescharacteristic, complemented
74C7×D5—151no—
75C7×D5—151no—
76C7×D5—452no—
77C32×C7—71no—
78C3×ΓL(1,𝔽8)C7×D2511yescharacteristic, complemented
79D25—31noHall
80C72C3×D7511yescharacteristic, complemented, Sylow
81C32×C5—71no—
82C2×C3×C7—5251no—
83D3×C7—251no—
84D3×C7—251no—
85D3×C7—752no—
86C2×ΓL(1,𝔽8)—751no—
87C5×C7—32no—
88C5×C7ΓL(1,𝔽8)×D1511yescharacteristic
89C5×C7C3×C7×D1511yescharacteristic
90D15—51no—
91C3×D5—1051no—
92C25D3×C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
93C3×C7—32no—
94C3×C7—71no—
95C3×C7—141no—
96C3×C7ΓL(1,𝔽8)×D2511yescharacteristic, complemented
97C3×C7C3×C7×D2511yescharacteristic, complemented
98ΓL(1,𝔽8)—21no—
99ΓL(1,𝔽8)C7×D7511yescharacteristic, direct product factor
100C3×D3—1751noHall
101C3×C5—71no—
102C3×C5—141no—
103C3×C5C7×D5×ΓL(1,𝔽8)11yescharacteristic
104C2×C7—751no—
105C2×C7—751no—
106C2×C7—2252no—
107D5—151no—
108C32—71noSylow
109C7—32no—
110C7ΓL(1,𝔽8)×D7511yescharacteristic, direct product factor
111C7C3×C7×D7511yescharacteristic, complemented
112C2×C3—5251no—
113D3—251no—
114C5C7×ΓL(1,𝔽8)×D1511yescharacteristic
115C3—71no—
116C3—141no—
117C3C7×ΓL(1,𝔽8)×D2511yescharacteristic, complemented
118C2—751noSylow
subgroupG/H
center, hypercenterC7ΓL(1,𝔽8)×D75
commutator, cosocle, hypocenterC3×C7×C25C2×C3×C7
FrattiniC5C7×ΓL(1,𝔽8)×D15
FittingC3×C72×C25C2×C3
socleC3×C5×C72C3×D5
LCSG ▷ C3×C7×C25
UCSC1 ◁ C7

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×C7×ΓL(1,𝔽8)×C25C2semidirect
2. C72×D75C3semidirect
6. C7×ΓL(1,𝔽8)×C25D3semidirect
7. C3×C72×C25C2×C3semidirect
9. ΓL(1,𝔽8)×D75C7direct (N × H)
13. C3×ΓL(1,𝔽8)×C25C2×C7semidirect
16. C72×C25C3×D3semidirect
18. C7×D75ΓL(1,𝔽8)direct (N × H)
19. C7×D75C3×C7semidirect
31. C3×C7×C25C2×ΓL(1,𝔽8)semidirect
32. C3×C7×C25C2×C3×C7semidirect
34. ΓL(1,𝔽8)×C25D3×C7semidirect
37. C3×C7×ΓL(1,𝔽8)D25semidirect
53. C7×C25D3×ΓL(1,𝔽8)semidirect
54. C7×C25C3×D3×C7semidirect
55. D75C7×ΓL(1,𝔽8)direct (N × H)
57. C3×C72C3×D25semidirect
59. C7×ΓL(1,𝔽8)D75direct (N × H)
73. C3×C25C2×C7×ΓL(1,𝔽8)semidirect
78. C3×ΓL(1,𝔽8)C7×D25semidirect
80. C72C3×D75semidirect
92. C25D3×C7×ΓL(1,𝔽8)semidirect
96. C3×C7ΓL(1,𝔽8)×D25semidirect
97. C3×C7C3×C7×D25semidirect
99. ΓL(1,𝔽8)C7×D75direct (N × H)
110. C7ΓL(1,𝔽8)×D75direct (N × H)
111. C7C3×C7×D75semidirect
117. C3C7×ΓL(1,𝔽8)×D25semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×C7×ΓL(1,𝔽8)×C252
C72×D753
C7×ΓL(1,𝔽8)×D253
C7×ΓL(1,𝔽8)×D155
C3×C7×D757
ΓL(1,𝔽8)×D757